Higher Spin Structures on Orbifolds

This project is concerned with spaces of m-spin structures on hyperbolic Riemann and Klein 2-orbifolds. A Riemann 2-orbifold is a 2-orbifold with a maximal atlas whose transition maps are holomorphic. It can be described as a quotient of the hyperbolic plane by a Fuchsian group. A Klein 2-orbifold is a 2-orbifold with a maximal atlas whose transition maps are either holomorphic or anti-holomorphic. It can be described as a hyperbolic Riemann 2-orbifold equipped with an anti-holomorphic involution. An m-spin structure on a hyperbolic Riemann 2-orbifold is a complex line bundle such that the m-th tensor power of this line bundle is isomorphic to the cotangent bundle of the orbifolds. To any m-spin structure on a Riemann 2-orbifold, one can assign a unique function on the space of homotopy classes of simple contours with values in Z/mZ, the associated m-Arf function. An m-spin structure on a hyperbolic Klein orbifold is an m-spin structure on the underlying Riemann 2-orbifold which is invariant under the anti-holomorphic involution.

Natanzon and Pratoussevitch used topological methods to study the space of m-spin structures on hyperbolic Riemann and Klein 2-orbifolds without cone points, to determine existence conditions and describe the topology of all connected components of the space of m-spin structures. Riley extended these results some cases of m-spin structures on hyperbolic Klein 2-orbifolds such that all cone points are outside of the fixed point set of the ant-holomorphic involution.

The aims of this project:
1. Study the spaces of m-spin structures on hyperbolic Klein 2-orbifolds, including the case of 2-orbifolds with cone points on the fixed point set of the anti-holomorphic involution.
2. Study some special families of m-spin structures on hyperbolic 2-orbifolds (hyperelliptic, polygonal, etc.) and their connections with branching points of the covering of the space of hyperbolic 2-orbifolds by the space of m-spin structures.
3. Investigate the connections with families of quasi-homogeneous Gorenstein singularities.